We have conducted a NanoSIMS ion imaging survey of about 1800 presolar silicon carbide (SiC) grains from the Murchison meteorite. A total of 21 supernova (SN) X grains, two SN C grains, and two putative nova grains were identified. Six particularly interesting grains, two X and C grains each and the two putative nova grains were subsequently studied in greater detail, namely, for C-, N-, Mg-Al-, Si-, S-, and Ca-Ti-isotopic compositions and for the initial presence of radioactive 26Al (half life 716000 yr), 32Si (half life 153 yr), and 44Ti (half life 60 yr). Their isotope data along with those of three X grains from the literature were compared with model predictions for 15 M⊙ and 25 M⊙ Type II supernovae (SNe). The best fits were found for 25 M⊙ SN models that consider for the He shell the temperature and density of a 15 M⊙ SN and ingestion of H into the He shell before the explosion. In these models a C- and Si-rich zone forms at the bottom of the He burning zone (C/Si zone). The region above the C/Si zone is termed the O/nova zone and exhibits the isotopic fingerprints of explosive H burning. Satisfactory fits of measured C-, N-, and Si-isotopic compositions and of 26Al/27Al ratios require small-scale mixing of matter originating from a region extending over 0.2 M⊙ for X and C grains and over 0.4 M⊙ for one of the putative nova grains, involving matter from a thin Si-rich layer slightly below the C/Si zone, the C/Si zone, and the O/nova zone. Simultaneous fitting of 14N/15N and 26Al/27Al requires a C-N fractionation of a factor of 50 during SiC condensation. This leads to preferential incorporation of radioactive 14C (half life 5700 yr) over directly produced 14N and can account for the 14N/15N along with 26Al/27Al ratios as observed in the SiC grains. The good fit for one of the putative nova grains along with its high 26Al/27Al points towards a SN origin and supports previous suggestions that some grains classified as nova grains might be from SNe. Apparent problems with the small-scale mixing scheme considered here are C/O ratios that are mostly <1 if C-, N-, and Si-isotopic compositions and 26Al/27Al ratios are simultaneously matched, underproduction of 32Si, and overproduction of 44Ti. This confirms the limitations of one-dimensional hydrodynamical models for H ingestion and stresses the need to better study the convective-boundary mixing mechanisms at the bottom of the convective He shell in massive star progenitors. This is crucial to define the effective size of the C/Si zone formed by the SN shock. The comparison between the Si isotope data of the SN grains and the models gives a hint that the predicted 30Si is too high at the bottom of the He burning shell.